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{\bf Stochastic analysis, spring  2013, Exercises-11, 18.04.2013 }


\begin{enumerate}
\item(Lenglart's inequality)

Let $X_t(\omega)\ge 0$ with $X_0=0$, and $A_t(\omega)\ge 0$  continuous processes adapted  with respect to  $\F=({\mathcal F}_t:t\in \R^+)$,
and assume that $A_t$ is non-decreasing such that for all {\bf bounded } stopping times $\tau(\omega)$
\begin{eqnarray} \label{ass:1}
E( X_{\tau} ) \le E( A_{\tau} ) 
\end{eqnarray}

We introduce the running maximum $X^*_t(\omega)= \max\limits_{0\le s \le t } X_s(\omega)$ 

Prove the following inequalities for {\bf all } $\F$-stopping times $\tau$, (also unbounded ):  $\forall \varepsilon , \delta > 0 $
\begin{eqnarray*} 
{\bf a ) } &&
P( X^*_{\tau} > \varepsilon ) \le \frac{ E( A_{\tau} ) }{  \varepsilon }
\\{\bf b ) } & &
P( X^*_{\tau} > \varepsilon , A_{\tau} \le \delta ) \le \frac{ E( A_{\tau} \wedge \delta  ) }{  \varepsilon }
\\{\bf c ) } & &
P( X^*_{\tau} > \varepsilon ) \le \frac{ E(  A_{\tau}  \wedge\delta) }{  \varepsilon } + P( A_{\tau} > \delta)
\end{eqnarray*}
Hint: First assume that $\tau$ is  a  bounded stopping time,  then you can 
use monotone convergence for unbounded stopping times.
 
Define
\begin{align*}
\sigma( \omega) = \inf\bigl\{ t:  X_t(\omega) > \varepsilon \}  
\end{align*}
and note that
\begin{align*}
\{ X^*_{\tau} > \varepsilon \} = \{  \sigma  <  \tau \} 
\end{align*}
Use the assumption (\ref{ass:1}) for the stopping time $\sigma \wedge \tau$.


\item
 Let $M_t$ a continuous $\F$-local martingale. The $\F$-predictable variation $\langle M \rangle_t$ is the non-decreasing process 
with  $\langle M \rangle_0=0$ such that
\begin{eqnarray*} 
 N_t=  M_t^2 - \langle M \rangle_t
\end{eqnarray*}
is a $\F$-local martingale.





Show that for any $\F$-stopping time $\tau$
\begin{eqnarray*} 
P\biggl(  \max\limits_{0\le s \le \tau(\omega) }  | M_s(\omega) | 
 > \varepsilon \biggr ) \le \frac{ E( \delta \wedge \langle M \rangle_{\tau} ) }{  \varepsilon^2 } + P( \langle M \rangle_{\tau} > \delta)
\end{eqnarray*}

Hint: use the previous exercise. In order to show that assumption (\ref{ass:1}) is satisfied, take a localizing
sequence  for $N_t$ and use Fatou lemma.

\item $\{ M^{(n)}_t(\omega)  \}_{n\in \N}$ a sequence of $\F$-local martingales and $\tau$ a $\F$-stopping time.
Show that  as $n\longrightarrow \infty$
\begin{eqnarray*} 
 \langle M^{(n)} \rangle_{\tau} \stackrel{P}{\to} 0 \quad \Longrightarrow \quad\max\limits_{0 \le s \le \tau }| M_{s}^{(n)}(\omega) | \stackrel{P}{\to} 0
\end{eqnarray*} 
with convergence in probability.


\item   Let $(B_t : t\ge 0)$ a Brownian motion in the filtration $\F=({\mathcal F}_t : t\ge 0)$.

\begin{enumerate}
\item Use Ito formula to show that 
\begin{align} 
Z_t(\theta) = \exp\biggl( \theta B_t -  \frac{\theta^2 } 2  t \biggr)
\end{align}
is a true martingale. 

Hint: We have shown this already by using the indpendence of the increments from the past.
here you are have to check the square integrability condition of an Ito integral.


\item
Use Ito's formula to compute the semimartingale decomposition of  $B_t^n$  for $n\in \N$. 
into a  local martingale and  a process of finite variation. Show that the local martingale is 
a true martingale. Recall that a Gaussian random variable is in $L^p(\Omega)$ $\forall p< \infty$.

Hint: use Ito's formula and verify the square integrability condition for  an Ito integral.


\item Compute $E_P( B_t^n )$, for $n\in \N$,by taking expectation in the semimartingale decomposition.
\end{enumerate}

\item
The Hermite polynomials are defined by the Taylor expansion
\begin{align}
F(x,u)=\exp\bigl( u x - \frac{ u^2}  2 \bigr) = \sum_{n=0}^{\infty} \frac { u^n}{ n! }  h_n(x) 
\end{align}
We see that $h_0(x)=1$.
We rewrite 
\begin{align*}
F(x,u)=
\exp\biggl( \frac{x^2}2 - \frac { (x-u)^2 }2\biggr) =
 e^{x^2/2 } \sum_{n=0}^{\infty} \frac{ u^n}{n!}
 \frac{ \partial^n }{\partial u^n} \exp\biggl( - \frac{ (t-u)^2} 2 \biggr) \bigg \vert_{u=0}
\end{align*}
which shows that
\begin{align*}
  h_n(x) =  (-1)^n \exp\biggl( \frac{ x^2} 2 \biggr) \frac{ d^ n }{ dx^n} \exp\biggl( \frac{ x^2} 2 \biggr)  , \quad n \ge 0 
\end{align*}

We also have
\begin{align*} 
 &
F\bigl( \frac x{\sqrt a}  , u \sqrt a \bigr)=
\exp\biggl(  u \sqrt{ a} \frac x {\sqrt  a} - \frac{ u^2 a} 2 \biggr)  & \\ &
=\exp\biggl(  u x - \frac{ u^2 a}  2 \biggr) =
 \sum_{n=0}^{\infty} \frac { u^n}{ n! } a^{n/2} h_n\bigl( \frac x {\sqrt a } \bigr) = \sum_{n=0}^{\infty} \frac { u^n}{ n! } H_n( x,a)
\end{align*}
with 
\begin{align} 
 H_n(x,a): = a^{n/2} h_n\bigl( \frac x {\sqrt a } \bigr).
\end{align}
We also set $H_n(x,0)=x^n$.

\begin{enumerate}

\item
Show that 
\begin{align*}
  \frac 1 2 \frac {\partial^2 }{ \partial x^2} H_n(x,a) +   \frac {\partial }{ \partial a} H_n(x,a) =0 
\end{align*}
Hint: use the Taylor expansion above.

\item Use Ito formula to  show when $B_t$ is an $\F$-Brownian motion, 
\begin{align*} 
 H_n( B_t,t ) = t^{n/2} h_n\bigl( B_t/ \sqrt t \bigr)
\end{align*}
is a martingale.  Justify the martingale property of the Ito integral.


\item Use Ito formula to show that when $M_t$ is a continuous local martingale in the $\F$-filtration,
\begin{align*}
H_n( M_t, \langle M \rangle_t ) =  \langle M \rangle_t^{n/2} h_n\biggl( \frac{ M_t}{ \sqrt { \langle M \rangle_t } } \biggr) 
\end{align*}  is a local martingale.

\item Show also that 
\begin{align*}
 H_n( M_t, \langle M \rangle_t ) = n! \int_0^t  \biggl( \int_0^{t_1} \dots \int_0^{t_{n-1}} dM_{t_n} dM_{t_{n-1} } \dots  \biggr) d M_{t_1}
\end{align*}
where on the right we have an {\bf iterated Ito integral }.





\end{enumerate}







\end{enumerate}





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